The codegree of a lattice polytope is the smallest integer dilate that contains a lattice
point in the relative interior. The weak maximum likelihood threshold of a statistical
model is the smallest number of data points for which there is a nonzero probability
that the maximum likelihood estimate exists. The codegree of a marginal polytope
is a lower bound on the maximum likelihood threshold of the associated
log-linear model, and they are equal when the marginal polytope is normal. We
prove a lower bound on the codegree in the case of hierarchical log-linear
models and provide a conjectural formula for the codegree in general. As an
application, we study when the marginal polytopes of hierarchical models are
Gorenstein, including a classification of Gorenstein decomposable models,
and a conjectural classification of Gorenstein binary hierarchical models.
